Inference

• Don't just fit models for the sake of it!
• What are our questions?
  • Relationship to covariates
  • Abundance
  • Distribution
  • Response to disturbance
  • Temporal changes
  • Other stuff?

• Evaluate the model, at a particular covariate combination
• Answering (e.g.) the question “at a given depth, how many dolphins?”
• Steps:
  1. evaluate the \( s(\ldots) \) terms
  2. move to the response scale (exponentiate? Do nothing?)
  3. (multiply any offset etc)

• in maths:
  • Model: \( \text{count}_i = A_i \exp \left( \beta_0 + s(x_i, y_i) + s(\text{Depth}_i)\right) \)
  • Drop in the values of \( x, y, \text{Depth} \) (and \( A \))
• in R:
  • build a data.frame with \( x, y, \text{Depth}, A \)
  • use predict()

preds <- predict(my_model, newdat=my_data, type="response")

(se.fit=TRUE gives a standard error for each prediction)

• Evaluate the fitted model at a given point
• Can evaluate many at once (data.frame)
• Don't forget the type=... argument!
• Obtain per-prediction standard error with se.fit

• \( \boldsymbol{\beta} \): uncertainty in the spline parameters

• \( \boldsymbol{\lambda} \): uncertainty in the smoothing parameter

• (Traditionally we've only addressed the former)

• (New tools let us address the latter…)

From theory:

\[ \boldsymbol{\beta} \sim N(\hat{\boldsymbol{\beta}}, \mathbf{V}_\boldsymbol{\beta}) \]

(caveat: the normality is only approximate for non-normal response)

What does this mean? Variance for each parameter.

In mgcv: vcov(model) returns \( \mathbf{V}_\boldsymbol{\beta} \).

• confidence intervals in plot
• standard errors using se.fit
• derived quantities? (see bibliography)

For regular predictions:

\[ \hat{\boldsymbol{\eta}}_p = L_p \hat{\boldsymbol{\beta}} \]

form \( L_p \) using the prediction data, evaluating basis functions as we go.

(Need to apply the link function to \( \hat{\boldsymbol{\eta}}_p \))

But the \( L_p \) fun doesn't stop there…

To get variance on the scale of the linear predictor:

\[ V_{\hat{\boldsymbol{\eta}}} = L_p^\text{T} V_\hat{\boldsymbol{\beta}} L_p \]

pre-/post-multiplication shifts the variance matrix from parameter space to linear predictor-space.

(Can then pre-/post-multiply by derivatives of the link to put variance on response scale)

• Recent work by Simon Wood
• “smoothing parameter uncertainty corrected” version of \( V_\hat{\boldsymbol{\beta}} \)
• In a fitted model, we have:
  • $Vp what we got with vcov
  • $Vc the corrected version

• Everything comes from variance of parameters
• Need to re-project/scale them to get the quantities we need
• mgcv does most of the hard work for us
• Fancy stuff possible with a little maths
• Can include uncertainty in the smoothing parameter too

• predict is your friend
• Most stuff comes down to matrix algebra, that mgcv sheilds you from
  • To do fancy stuff, get inside the matrices